That 500 w power supply will be perfectly fine to test with, especially if you do it the "best" way (see below).
But it won't hurt to try and boot with the smaller power supply before you try the "big test" below. Windows might complain, but the boot process will go through if that is the problem.
The best way to diagnose a non-booting PC:
- Disconnect everything (all cables, drives, etc) from the rear of the PC, and the motherboard.
- Remove all cards, including the video card.
- Remove all the RAM.
You are now left with just 3 components: power supply (500w) connected, motherboard, and CPU. For an even more thorough test a lot of folks remove the motherboard and place it outside of the case.
Then plug in or turn on the power supply and start the computer.
You won't see anything, of course, but you should get some error beeps. This is good. They are telling you that the boot process is proceeding normally and "no RAM is found".
If the MB is out of the case, now is the time to shut down and put it back in. Watch those stand-offs! Then boot again to retest for the "no RAM" error.
Power down and then install one stick of RAM in slot A1. Boot again. More (different pattern) beeps mean "no video card detected". Good. This means everything is working as it should.
If you don't get the BIOS error beeps at any point then you have found the problem hardware.
Power down and then install the video card, and connect the monitor. Boot again. There should be no error beeps at this point and you should see the BIOS screens on the monitor. It will stop when it gets to "no boot device found".
Now just keep adding components back to the system, one at a time, to test to see if the symptoms return. Start with the remainder of the RAM - one stick at a time.
Hope that helps.